## Leonhard Euler

## Propositional Logic by Ali Nesin

Kaan Asker

Introduction

Hello, this week we will talk about Ali Nesin’s book Propositional Logic (Önermeler Mantığı). This book is his first book in his Introduction to Mathematics series. In the book, there is a wide range of topics, from pure logic to mathematical induction. In this review, a brief summary of the sections of the book will be given, and the book will be evaluated.

Ali Nesin suggests this book is for people who are interested in logic, mathematical logic, and mathematics. He also thinks that the book is designed for a high school student so that he/she understands the first 9 sections of the book. He says: “Everyone who wants to introduce himself/herself to ‘real’ mathematics will find this book useful.” And as a joke, he says: “I don’t recommend this book to ones who think learning logic will make himself/herself more logical.”

What is a Proposition?

Propositions in daily usage mean a statement that can be true or false such as 2 + 2 = 5 or 2 + 2 = 4, but in propositional logic, the trueness or falsehood of statements is not inspected; rather, the relationship between statements are evaluated. For example, in propositional logic, individual statements are assumed to be true, such as:

If p or not p

If p, then p

If p and q, then p

If p, then p or q

These statements are always true, even if p and q are wrong. So, propositional logic evaluates the trueness or falsehood of the relationship between statements.

If there are two pencils in my backpack, then Fatma is a Mathematician.

Let us evaluate this proposition. Note that this statement consists of two statements. Let the first statement, If there are two pencils in my backpack, be p and the second statement, Fatma is a Mathematician, be q. The conjunction “then” is symbolized as an arrow, →. Therefore our statement is,

p → q

Let’s draw a chart for the trueness or the falsehood of the statements.

We will place true or false in the places of the question marks. The last row means if there are two pencils in the backpack, Fatma is a mathematician. If both of the statements, p and q are true, then the proposition p → q will be true as can be easily understood by common sense. Also, if there are two pencils, but in conclusion, Fatma is not a mathematician (row 3), the statement will be false. Therefore:

However, I shall warn you, the remaining rows are not as easy as the initial ones. If the first statement is wrong, then we will assume the proposition is true.

The reason for this is not clear as the cases where p was true. If I don’t have two pencils in my backpack, then the statement is true regardless of Fatma being a mathematician or not.

Ali Nesin stresses that these two last statements are not that important since if the first statement is false, the second statement is not important. This is because the proposition will not give us any information. The statement has a value if and only if the first statement is true because then it is giving information.

Although one could argue that p → q is not necessarily true when p is false, if something can be either true or wrong, we should choose the true one. Think of the statement

If p and p → q, then q

"And" is represented by Ʌ, so in other words:

(p Ʌ (p → q)) → q

We want this proposition to be true regardless of whether p is true or false. If when p was false, p → q was also false, then, the proposition (p Ʌ (p → q)) → q would also be false because of the statement p being false. This is not what we aim for, so when p is false, p → q must be inevitably true. In conclusion, the trueness chart for p → q is:

As can be seen from the chart, p → q is false if and only if p is true and q is false; otherwise, it is true.

Mathematical Induction

The second most important part of the book was the topic of mathematical induction, a very powerful weapon in mathematics. Unlike natural-scientific induction , mathematical induction is without flaws. Mathematical induction is a 3-phase process which works for proving mathematical theorems. If you are taking IB, you will learn this topic in Maths HL class in the 11th grade anyway.

In mathematical induction, we prove the theorem first for the number 1. Then, we assume the theorem is true for the number n. Afterwards, we show that the theorem is true for the number n+1 if it is true for n. Therefore, we understand that the theorem is true for all natural numbers because the theorem is true for 1, and because it is true for 2, it is true also for 3. This will continue till infinity, but strictly for natural numbers (0, 1, 2, 3, 4 …).

Let’s prove the theorem:

The sum of all natural numbers’ squares from 1 to n is

1. Step: Prove for n=1

Therefore the statement is true for n=1.

2. Step: Assume the trueness of n=n

If we assume this statement true and predict the n+1 case if the n is true,

3. Step: Prove for n = n+1

As it can be seen, Equations 1 and 2 are equal; therefore, the theorem is proved for all natural numbers. You can substitute any number; try the sum of squares till 129058, and you will see it works.

Mathematical induction is the reason why Ali Nesin fell in love with mathematics and planned to be a mathematician in the future when he was 14.

Evaluation of the Book

This book is not a serious textbook, but it contains a lot of mathematical knowledge. It consists of 160 pages, and you must have a pencil and a blank page in your reach to understand the contents while reading it. This book is not for reading when lying in bed or sunbathing in Bodrum. The book has to be studied in a disciplined fashion.

The book contains many jokes, and it is sometimes informal. For example, when explaining the propositional logic in “if and then” statements, Ali Nesin narrates the story of Bertrand Russel claiming himself to be the pope according to the false principle 1=0. Russel says to add 1 to each side, get 2=1. Then, close the pope and me to an empty room. How many people are there in this room? The audience replies 2. Bertrand Russel says, since 2=1, the pope is me, I am the pope. As Russel demonstrates, false mathematics can lead an atheist to become the pope.

Conclusion

Well, I recommend this book because of its simplicity and because of its importance in learning mathematics. Maths is built upon the ideas of propositional logic; therefore, anyone who does not know the basics of logic would not comprehend mathematics to a full extent.

1 You can read more about the Problem of Induction if you are interested in why induction does not work in natural sciences.